, 2019

When we say color in this solution we mean red, blue, yellow and green.
An $n$-tuple $(c_1, c_2, \dots, c_n)$ of colors (in which the same color can appear any number of times) is called consistent if every color appearing appears an even number of times, or every color appearing appears an odd number of times. Note that $n$ must be even if $(c_1, c_2, \dots, c_n)$ is consistent. The conditions imposed for the problem is equivalent to the statement that for any row and any column the colors $(c_1, c_2, c_3, c_4)$ used to paint is consistent.
Lemma: Let $n$ be an odd number, and let $(c_1, c_2, \dots, c_n)$ be an $n$-tuple of colors. Then, there exists a unique color $d$ for which $(c_1, c_2, \dots, c_n, d)$ becomes consistent.
Proof: Since $n$ is odd, there are odd number of colors which appear odd number of times among $c_1, \dots, c_n$. Thus, there is one color for which the even-odd parity of the appearances among $c_1, \dots, c_n$ is different from other colors appearing there, and this color is $d$.
Let us consider how many ways there are for coloring squares of the given grid to satisfy the conditions of the problem. As indicated in the diagram below, we indicate colors by using symbols. There are $4^9$ ways of choosing colors if we choose $c_1, c_2, \dots, c_9$ freely. If we fix one such choice, then by applying the Lemma to the $3$ upper rows to obtain $d_1, d_2, d_3$, and by applying the Lemma to $3$ left columns to obtain $e_1, e_2, e_3$. If we can show that the color $x$ for which $(d_1, d_2, d_3, x)$ is consistent coincides with the color $y$ for which $(e_1, e_2, e_3, y)$ is consistent, then this common color must be the unique color that should be used to color the right lowest corner of the diagram below.
Since each one of the quadruples
$$(c_1, c_2, c_3, d_1), (c_4, c_5, c_6, d_2), (c_7, c_8, c_9, d_3), (d_1, d_2, d_3, x)$$
is consistent, the $16$-tuple obtained by combining these four quadruples is consistent, and so is the $10$-tuple obtained by deleting each of $(d_1, d_2, d_3)$ twice from the $16$-tuple above. Thus, we conclude that $(c_1, c_2, \dots, c_9, x)$ is consistent. Arguing in the same way, we also get the fact that $(c_1, c_2, \dots, c_9, y)$ is consistent. In view of the Lemma above, we conclude that $x$ and $y$ represent the same color.
No matter how we choose the colors $c_1, c_2, \dots, c_9$, we see that there is a unique choice of the color to color the right lowermost square of the diagram below. So, $4^9 = 262144$ is the desired answer to the problem.